Download Statistics 1: Problem Set 3 r.v. . random variable cdf . cumulative

Statistics 1: Problem Set 3
r.v. = random variable
cdf = cumulative distribution function [
Abbreviations:
or distribution function]
pmf = probability mass function
pdf = probability density
function
1. Consider the experiment of throwing a fair die. Let X be the r.v. which assigns 1 if
the number that appears is even and 0 if the number that appears is odd. (a) What is the
range of X ? (b) Find P(X = 1) and P(X = 0).
2. Consider the experiment of tossing a coin three times. Let X be the r.v. giving the
number of heads obtained. We assume that the tosses are independent and the probability
of a head is p. (a) What is the range of X ? (b) Find the probabilities (i) P(X = 0); (ii)
P(X = 1); (iii) P(X = 2); and (iv) P(X = 3). (c) Find (i) P(X 1); (ii) P(X > 1); and (iii)
P(0 < X < 3).
3. Consider the function given by
F (x) =
8
<
0 if x < 0
x + 1=2 if 0 x < 1=2
:
1 if x 1=2
(a) Sketch F(x) and show that F(x) has the properties of a cdf. (b) If X is the r.v.
whose cdf is given by F(x), …nd (i) P(X 1=4)(ii), P( 0 < X 1=4)(iii) P (X = 0), and (iv)
P(0 X 1=4).
4. Suppose a discrete r.v. X has the following pmfs:
Px (1) = 4
Px (2) = 1=4
Px (3) = 1=8
Px (4) = 1=8
(a) Find and sketch the cdf Fx (x) of the r.v. X. (b) Find (i) P(X
(iii) P(1 X 3).
5. (a) Verify that the function p(x) de…ned by
x
p(x) =
3=4(1=4)
if x = 0; 1; 2; : : :
0 otherwise
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1), (ii) P(1 < X
3),
is a pmf of a discrete r.v. X. (b) Find (i) P(X = 2), (ii) P(X
2), (iii) P(X
1).
6. The pdf of a continuous r.v. X is given by
8
< 1=3 if 0 < x < 1
2=3 if 1 < x < 2
fx (x) =
:
0 otherwise
Find the corresponding cdf Fx (x) and sketch fx (x) and Fx (x).
7. Let X be a continuous r.v. X with pdf
fx (x) =
kx if 0 < x < 1
0 otherwise
where k is a constant. (a) Determine the value of k and sketch fx (x). (b) Find and sketch
the corresponding cdf Fx (x). (c) Find P(1=4 < X 2).
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